Γ-convergence for nonlocal phase transitions

We discuss the Γ-convergence, under the appropriate scaling, of the energy functional∥u∥ Hs(Ω) 2+∫ ΩW(u)dx, with s∈(0,1), where ∥u∥H s (Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. When s∈[1/2,1), we show that the energy Γ-converges to the classical minimal surface functional - while, when s∈(0,1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.